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<div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">#!/usr/bin/python</span>
<a name="l00002"></a>00002 <span class="comment"># -*- coding: utf-8 -*-</span>
<a name="l00003"></a>00003 
<a name="l00004"></a>00004 <span class="comment"># Copyright (c) 2011</span>
<a name="l00005"></a>00005 <span class="comment">#</span>
<a name="l00006"></a>00006 <span class="comment"># Permission is hereby granted, free of charge, to any person obtaining a</span>
<a name="l00007"></a>00007 <span class="comment"># copy of this software and associated documentation files (the &quot;Software&quot;),</span>
<a name="l00008"></a>00008 <span class="comment"># to deal in the Software without restriction, including without limitation</span>
<a name="l00009"></a>00009 <span class="comment"># the rights to use, copy, modify, merge, publish, distribute, sublicense,</span>
<a name="l00010"></a>00010 <span class="comment"># and/or sell copies of the Software, and to permit persons to whom the</span>
<a name="l00011"></a>00011 <span class="comment"># Software is furnished to do so, subject to the following conditions:</span>
<a name="l00012"></a>00012 <span class="comment">#</span>
<a name="l00013"></a>00013 <span class="comment"># The above copyright notice and this permission notice shall be included in</span>
<a name="l00014"></a>00014 <span class="comment"># all copies or substantial portions of the Software.</span>
<a name="l00015"></a>00015 <span class="comment">#</span>
<a name="l00016"></a>00016 <span class="comment"># THE SOFTWARE IS PROVIDED &quot;AS IS&quot;, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR</span>
<a name="l00017"></a>00017 <span class="comment"># IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,</span>
<a name="l00018"></a>00018 <span class="comment"># FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE</span>
<a name="l00019"></a>00019 <span class="comment"># AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER</span>
<a name="l00020"></a>00020 <span class="comment"># LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,</span>
<a name="l00021"></a>00021 <span class="comment"># OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE</span>
<a name="l00022"></a>00022 <span class="comment"># SOFTWARE.</span>
<a name="l00023"></a>00023 <span class="comment">#</span>
<a name="l00024"></a>00024 <span class="comment"># Author: Jesus Carrero &lt;j.o.carrero@gmail.com&gt;</span>
<a name="l00025"></a>00025 <span class="comment">#</span>
<a name="l00026"></a>00026 <span class="comment"># Look at this page for using diag sparse matrices in scipy:</span>
<a name="l00027"></a>00027 <span class="comment"># http://pod.tamu.edu/~lmkli/reference/python/scipy_html/generated/scipy.sparse.spdiags.html</span>
<a name="l00028"></a>00028 <span class="comment">#</span>
<a name="l00029"></a>00029 <span class="comment"># On Cubic Splines and the implementation here:</span>
<a name="l00030"></a>00030 <span class="comment"># Numerical Algorithms in C.</span>
<a name="l00031"></a>00031 <span class="comment"># [Gisela Engeln-Mullges and Frank Uhligi.] (1996) Chapter 10</span>
<a name="l00032"></a>00032 
<a name="l00033"></a>00033 <span class="keyword">from</span> numpy <span class="keyword">import</span> zeros, ones, exp, linspace
<a name="l00034"></a>00034 <span class="keyword">from</span> numpy <span class="keyword">import</span> array, r_, c_, roll
<a name="l00035"></a>00035 <span class="keyword">from</span> scipy.sparse.construct <span class="keyword">import</span> spdiags
<a name="l00036"></a>00036 <span class="keyword">from</span> scipy.sparse.linalg <span class="keyword">import</span> spsolve <span class="keyword">as</span> spsolve
<a name="l00037"></a>00037 
<a name="l00038"></a>00038 
<a name="l00039"></a><a class="code" href="classutils_1_1Spline_1_1Spline.html">00039</a> <span class="keyword">class </span><a class="code" href="classutils_1_1Spline_1_1Spline.html">Spline</a>:
<a name="l00040"></a>00040 
<a name="l00041"></a>00041     <span class="stringliteral">&quot;&quot;&quot;</span>
<a name="l00042"></a>00042 <span class="stringliteral">        This is an implementation of natural cubic splines</span>
<a name="l00043"></a>00043 <span class="stringliteral">        Sp(x) = a + b * (x-xj) + c *(x-xj)^2 + d *(x-xj)^3</span>
<a name="l00044"></a>00044 <span class="stringliteral">    &quot;&quot;&quot;</span>
<a name="l00045"></a>00045 
<a name="l00046"></a>00046     __slots__ = [<span class="stringliteral">&#39;m_xi&#39;</span>, <span class="stringliteral">&#39;m_yi&#39;</span>, <span class="stringliteral">&#39;m_spc&#39;</span>]
<a name="l00047"></a>00047 
<a name="l00048"></a>00048     <span class="keyword">def </span>__init__(self, _xi=None, _yi=None):
<a name="l00049"></a>00049         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a> = _xi
<a name="l00050"></a>00050         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a106770b0ced2f116807daded1344337a">m_yi</a> = _yi
<a name="l00051"></a>00051         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a4accd7aae8d5d762768c033cbd95ae6f">m_spc</a> = <span class="keywordtype">None</span>
<a name="l00052"></a>00052         <span class="keywordflow">if</span> <span class="keywordflow">not</span> (_xi <span class="keywordflow">is</span> <span class="keywordtype">None</span> <span class="keywordflow">and</span> _yi <span class="keywordflow">is</span> <span class="keywordtype">None</span>):
<a name="l00053"></a>00053             self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#af2068f5be4a17e2c921a2f4d11c727a5">spline</a>(_xi, _yi)
<a name="l00054"></a>00054 
<a name="l00055"></a>00055     <span class="keyword">def </span>_compSplineCoeff(self):
<a name="l00056"></a>00056         (x, y) = (self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a>, self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a106770b0ced2f116807daded1344337a">m_yi</a>)
<a name="l00057"></a>00057 
<a name="l00058"></a>00058         one = ones((1, 1))
<a name="l00059"></a>00059         zer = zeros((1, 1))
<a name="l00060"></a>00060         dx = roll(x, -1) - x
<a name="l00061"></a>00061         dx = dx[:-1]  <span class="comment"># INTERVAL LENGTH</span>
<a name="l00062"></a>00062         d = 2 * (dx + roll(dx, -1))
<a name="l00063"></a>00063         d = r_[r_[one, d[1:]], one]
<a name="l00064"></a>00064         u = r_[zer, dx.copy()]
<a name="l00065"></a>00065         u[1] = 0
<a name="l00066"></a>00066         l = r_[dx.copy(), zer]
<a name="l00067"></a>00067         l[-2] = 0
<a name="l00068"></a>00068 
<a name="l00069"></a>00069         b = (roll(y, -1) - y)[:-1] / dx
<a name="l00070"></a>00070         b = 3. * (roll(b, -1) - b)
<a name="l00071"></a>00071         b = r_[r_[zer, b[:-1]], zer]
<a name="l00072"></a>00072 
<a name="l00073"></a>00073         data = array([list(d.flatten()), list(l.flatten()),
<a name="l00074"></a>00074                      list(u.flatten())])
<a name="l00075"></a>00075 
<a name="l00076"></a>00076         diags = array([0, -1, 1])
<a name="l00077"></a>00077         M = spdiags(data, diags, d.size, d.size).tocsc()
<a name="l00078"></a>00078         c = spsolve(M, b, use_umfpack=<span class="keyword">False</span>)
<a name="l00079"></a>00079         c = c.reshape((c.size, 1))
<a name="l00080"></a>00080 
<a name="l00081"></a>00081         <span class="comment"># SPLINE COEFFICIENTS</span>
<a name="l00082"></a>00082 
<a name="l00083"></a>00083         spc = zeros((x.size - 1, 4), <span class="stringliteral">&#39;float64&#39;</span>)
<a name="l00084"></a>00084         spc[:, 0] = y[:-1].flatten()  <span class="comment"># a</span>
<a name="l00085"></a>00085         spc[:, 1] = ((roll(y, -1) - y)[:-1] / dx - dx * (roll(c, -1)
<a name="l00086"></a>00086                      + 2 * c)[:-1] / 3.).flatten()  <span class="comment"># b</span>
<a name="l00087"></a>00087         spc[:, 2] = c[:-1].flatten()  <span class="comment"># c</span>
<a name="l00088"></a>00088         spc[:, 3] = 1. / 3. * ((roll(c, -1) - c)[:-1] / dx).flatten()  <span class="comment"># d</span>
<a name="l00089"></a>00089         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a4accd7aae8d5d762768c033cbd95ae6f">m_spc</a> = spc
<a name="l00090"></a>00090 
<a name="l00091"></a>00091     <span class="keyword">def </span>get_grid(self):
<a name="l00092"></a>00092         <span class="keywordflow">return</span> self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a>
<a name="l00093"></a>00093 
<a name="l00094"></a>00094     <span class="keyword">def </span>getFuncValue(self):
<a name="l00095"></a>00095         <span class="keywordflow">return</span> self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a106770b0ced2f116807daded1344337a">m_yi</a>
<a name="l00096"></a>00096 
<a name="l00097"></a>00097     <span class="keyword">def </span>spline(self, _xi, _yi):
<a name="l00098"></a>00098         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a> = _xi.reshape((_xi.size, 1))
<a name="l00099"></a>00099         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a106770b0ced2f116807daded1344337a">m_yi</a> = _yi.reshape((_yi.size, 1))
<a name="l00100"></a>00100         self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a12159befe1117c936b23900a7a4d5c2c">_compSplineCoeff</a>()
<a name="l00101"></a>00101 
<a name="l00102"></a>00102     <span class="keyword">def </span>value(self, x, i):
<a name="l00103"></a>00103         a = self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a4accd7aae8d5d762768c033cbd95ae6f">m_spc</a>[i, :]
<a name="l00104"></a>00104         dx = x - self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a>[i]
<a name="l00105"></a>00105         <span class="keywordflow">return</span> a[0] + dx * (a[1] + dx * (a[2] + dx * a[3]))
<a name="l00106"></a>00106 
<a name="l00107"></a>00107     <span class="keyword">def </span>derivative(self, x, i):
<a name="l00108"></a>00108         a = self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a4accd7aae8d5d762768c033cbd95ae6f">m_spc</a>[i, :]
<a name="l00109"></a>00109         dx = x - self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a>[i]
<a name="l00110"></a>00110         <span class="keywordflow">return</span> a[1] + dx * (2. * a[2] + dx * 3. * a[3])
<a name="l00111"></a>00111 
<a name="l00112"></a>00112     <span class="keyword">def </span>secondDerivative(self, x, i):
<a name="l00113"></a>00113         a = self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a4accd7aae8d5d762768c033cbd95ae6f">m_spc</a>[i, :]
<a name="l00114"></a>00114         dx = x - self.<a class="code" href="classutils_1_1Spline_1_1Spline.html#a1d2647a3a631580e6c0efbeb61e542f9">m_xi</a>[i]
<a name="l00115"></a>00115         <span class="keywordflow">return</span> 2. * a[2] + 6. * a[3] * dx
<a name="l00116"></a>00116 
<a name="l00117"></a>00117 
<a name="l00118"></a>00118 <span class="keyword">def </span>f(x):
<a name="l00119"></a>00119     <span class="keywordflow">return</span> 10 + 20 * x - 30 * x ** 2
<a name="l00120"></a>00120     <span class="keywordflow">return</span> x ** 3
<a name="l00121"></a>00121     <span class="keywordflow">return</span> ones(x.shape)
<a name="l00122"></a>00122 
<a name="l00123"></a>00123 
<a name="l00124"></a>00124 <span class="keywordflow">if</span> __name__ == <span class="stringliteral">&#39;__main__&#39;</span>:
<a name="l00125"></a>00125     nSeg = 100
<a name="l00126"></a>00126     x = linspace(0, nSeg, nSeg, <span class="keyword">False</span>)
<a name="l00127"></a>00127     y = f(x)
<a name="l00128"></a>00128     mySpline = <a class="code" href="classutils_1_1Spline_1_1Spline.html">Spline</a>(x, y)
<a name="l00129"></a>00129 
<a name="l00130"></a>00130     <span class="comment"># chk Results</span>
<a name="l00131"></a>00131 
<a name="l00132"></a>00132     shift = 0.5
<a name="l00133"></a>00133     yt = f(x + shift * ones(x.shape))
<a name="l00134"></a>00134     spVals = array([mySpline.value([x[s] + shift], s) <span class="keywordflow">for</span> s <span class="keywordflow">in</span>
<a name="l00135"></a>00135                    range(x.size - 1)]).flatten()
<a name="l00136"></a>00136     <span class="keywordflow">print</span> <span class="stringliteral">&#39; x, error &#39;</span>
<a name="l00137"></a>00137     <span class="keywordflow">print</span> c_[(x + shift)[:-1], yt[:-1] - spVals]
<a name="l00138"></a>00138 
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